The teaching of mathematics requires dedicated and talented teachers. The objective is to mold students in ways that will provide them with a good understanding of mathematics. A mathematics teacher must provide each student with the ability to reason and understand basic principles. The basic principles must be learned at an early stage, so the students can bring this knowledge forward to higher levels, and be successful in life. The tools and information that teachers bring into the classroom can greatly influence the results that they can achieve.
In developing the invention, insight was gathered from observations of teaching techniques used in 7th grade math, and the tools that could be purchased in the open market.
In a typical 7th grade class, math comprises the use of tools, such as: fraction bars, fraction circles, pattern blocks, cubes, and integer chips. These commercially available visuals can help students to understand fractions. There are limitations, however, with using these manipulative devices.
Desiring to add or subtract fractions utilizing these purchased visuals, one is limited by only enough pieces of each fractional size to make a whole. This restricts the types and number of problems one can teach. Construction paper is often used to cut the paper into shapes and sizes that assist in the teaching of mathematics. This technique, however, is very time consuming.
In addition, if a piece of any set of materials is lost, the set's usability is severely impaired.
These fractional visuals have an upper fraction limit of twelfths, and generally exclude sevenths.
Consider the problem ⅗+⅔. What would a student use for the common denominator using the commercial visuals? There are no fifteenths in a commercial set to solve the problem. Also, having all the possible piece sizes available beforehand would impair the reasoning process. This is so, because too many different sized pieces in each fraction could add confusion, and turn the finding of the common denominator into a trial and error process. Trial and error is only appropriate at early stages of the mathematics learning process. It is much more appropriate to let the process evolve to an understanding of the underlying essence of the operation of fractions.
Using fraction bars, and fraction circles to find a common denominator is limited, because only part of the fraction is presented.
Some kits that are currently commercially available have a whole cut into a form upon which you can lay the fractional part. One is able to observe the part of the whole that is not filled. However, one does not visualize the size of the pieces that are missing to make the whole. That is to say, the space is empty of any visual realization. A student may possibly grasp the problem, when the student trades the pieces for a smaller size. It is realized that one can substitute two little pieces for a larger piece. However, the student does not realize that the same thing is actually happening to the pieces that are missing. In other words, this commercial kit or system fails to provide the necessary reasoning needed to solve the problem.
Working with fractions, requires the ability to reason how to arrive at a size for the least common denominator that will accommodate both fraction parts. Teaching students how to reason is the proper way to teach mathematics.
In developing the present invention, it was realized that moldable materials provided a means by which fractions and other mathematical problems could be taught. Moldable materials seemed to solve all of the aforementioned concerns.
Some inspection of the commercially available moldable dough enabled students to use reasoning to determine the size for a common denominator. Students did not need to experiment. Students gained a lot of understanding using the dough. Many students developed positive associative reasoning using the dough. These positive feelings transferred to other mathematical problems. In other words, it was realized that moldable materials were the proper teaching vehicles to develop mathematical reasoning.
However, it was also realized that molding and manipulating dough or clay in classrooms had some serious disadvantages, viz., molding and manipulating the dough to the proper shapes and sizes was time consuming. This left preciously little time for teaching the overall mathematical process. The inventor tried to find a way to quickly size and shape the dough or clay.
It was determined that an extruder was needed for obtaining the proper size and shape of the moldable materials. Some commercial children's dough kits are utilized only as playing tools, and contain a bucket of moldable dough. The bucket comprises a rolling pin, and some shape cutters for making geometric and animal shapes. Also, the system comes with an extruder with three sliders having different shapes for extruding the dough. All the openings (die orifices) of the extruder slider were under five eights of an inch. The extruder could be adapted to make rectangles of ½ inch. One half inch, however, is not wide enough to accommodate students ranging over a variety of levels of motor skills, coordination, and dexterity. Rectangle-shaped cutters were not provided.
A master patternmaker was hired by this inventor to modify this commercial extruder, and make a new slider, in order to produce a slot (die orifice) capable of making a rectangle one and one half inches wide. Using this modified extruder and the new slider, this inventor was able to produce a rectangular prism that was six and three quarter inches long, one and one half inches wide, and about one-quarter to three-sixteenths thin. This rectangular solid was produced in just a few seconds using super soft children's dough.
The present invention comprises a kit for making parts for math processes. The kit includes a container for housing moldable material, such as: dough or clay. The kit also comprises an extruder, and extruder attachments, for varying the shape and/or size of the extruding die orifice to fit mathematical needs. Other cutting or shaping accessories can be included, along with a work mat. Forms can be included showing the “whole” in which the fraction can be deposited. The forms can be printed upon the work mat. This is a powerful advantage, because it assists students to visualize the abstract written steps, while performing mechanical steps with the dough. The mat surface should allow for magic marker and erasable writing.
A guide, or instruction booklet is also enclosed in order to assist the teacher in the use of the materials for teaching mathematical processes. A CD, DVD, or media disk is included to visually instruct both teacher and student. Support materials and instructions can also be provided via a web site. A workbook can be enclosed as part of the accessories. A mold or a grid can also be part of the kit.
The advantage of this invention over other systems disclosed in the art is its versatility and ease of use. Using moldable material, one can quickly extrude the desired common denominator pieces, and also the pieces that are missing from both the top number of your fraction. In cutting all the pieces to make a whole, and finding a common denominator, one is also cutting the pieces one has as well as the pieces that are missing. (That explains why you multiply both the top number of the fraction (the number of pieces you have) and the bottom number (the number of pieces it takes to make a whole—the total of the pieces one has and the missing pieces) by the same number—no matter how many times one cuts each piece.
Many teachers who use fraction bars or fraction circles for teaching addition and subtraction, switch to a drawn model to show multiplication and division. This is so, because they can show additional cuts in their fractions representing multiplication and division.
Multiplication and division are easy to show with fraction rectangles made of moldable material. Moldable materials can be used to represent one fraction with vertical cuts, and then horizontal cuts can be made to represent the multiplication of that fraction by another fraction. There is no need to change models. Students can see what happens directly, when division is performed. In division, moldable material can represent both fractions separately, and then cuts can be placed where necessary to show how many times one fraction fits evenly into another. This works fine even if the answer is less than one.
It should be understood, that the invention is not limited to teaching multiplication and division within fraction problems. This inventive kit can also teach percentages, algebra, geometry, and percentage problems.
It is clear that all four operations with fractions can be comfortably accomplished with moldable materials. This can give students a greatly increased capacity for numbers, operations, and quantitative reasoning. This is so, because using the same model consistently reinforces the connections they make between operations done with the model, and helps them identify similarities and differences between these operations. The preferred model shape is a rectangle, but other shapes such as circles can be used.
The important goal in teaching is not to show children how to get answers by repetitive techniques. Rather, the true goal seeks to provide students with an understanding of exactly what is happening in the physical operations that are performed. The teacher wants these techniques to become comfortable in the student's psyche, as one would gather comfort from old friends. The teacher desires that the mystery so often associated with performing mathematical tasks be eliminated. It is also very important to connect steps on the written page with the steps of the process being physically preformed with the model. Using moldable material allows for the application of processes to be associated with the model. The invention provides a means of clearly showing what is happening in each step on the written work page. The processes made affordable by the invention not only provide a wonderful tool for student understanding, but the invention also provides concrete evidence for the teacher, whether the student has grasped and understood the mathematical process.
In summation, the kit of this invention is designed to expand the advantages of moldable materials in the teaching of mathematics. This expansion is directly linked to a device or method for quickly achieving the necessary shapes and sizes of the desired moldable materials. The use of an extruder fits neatly into the need for quickly obtaining the desired shapes and sizes, and for providing the necessary quantities of the materials sought. Hereinafter, the proper size of extrusions will be discussed. Although the invention features a simple mechanical extruder, it is contemplated that battery powered extrusion tools, such as motor-driven augurs feeding material through a chosen die, can also be part of this invention.